Integration Overview – A Level Maths Year 2

In Year 1 you could integrate only powers of $x$ (see Year 1 Integration). Year 2 extends integration to every function you can now differentiate — exponentials, logarithms and trig functions — and gives you a toolkit of techniques for products, compositions and fractions.

Integration is reverse differentiation, and every technique on these pages is a differentiation rule run backwards: the chain rule becomes recognition and substitution, the product rule becomes integration by parts. Unlike differentiation, though, there is no single routine that always works — the skill is choosing the right method.

What is new compared with Year 1

$e^x$, $\dfrac{1}{x}$, $\sin x$, $\cos x$, $\sec^2 x$ · the reverse chain rule · trig identities
Substitution · integration by parts · partial fractions
Integration as the limit of a sum · areas between curves · parametric areas
First-order separable differential equations and modelling

Calculators are allowed, and many will evaluate a definite integral numerically. But questions that ask for an exact answer, or say "show detailed reasoning" or "use algebraic integration", need full working — a decimal from the calculator earns nothing. Use it to check your exact answer instead.

Word	Meaning	Example
Integrand	The function being integrated.	$x e^x$ in $\int x e^x\,dx$
Indefinite integral	A family of functions, with $+\,c$.	$\int\cos x\,dx = \sin x + c$
Definite integral	A number found using limits.	$\int_0^{\pi}\sin x\,dx = 2$
Limits	The values at the ends of the interval.	$0$ and $\pi$ above
Substitution	Changing the variable to simplify an integral.	$u = x^2 + 1$
By parts	The reverse of the product rule.	$\int u\,v'\,dx = uv - \int v\,u'\,dx$
Differential equation	An equation containing a derivative.	$\dfrac{dy}{dx} = ky$
Separable	Can be written $\dfrac{dy}{dx} = \mathrm{f}(x)\mathrm{g}(y)$.	$\dfrac{dy}{dx} = xy^2$
General / particular solution	With an unknown constant / with the constant found.	$y = Ae^{3x}$ / $y = 5e^{3x}$

Radians only. Every trigonometric integral (and derivative) result assumes $x$ is in radians. Set your calculator to radians before evaluating any definite trig integral.

Work down this list. Stop at the first one that fits.

1 — Is it a standard result, possibly with a linear inside?

$\int e^{ax+b}\,dx = \tfrac{1}{a}e^{ax+b} + c$ · $\int\dfrac{1}{ax+b}\,dx = \tfrac{1}{a}\ln|ax+b| + c$

2 — Can it be rewritten? (expand, split, use an identity)

$\sin^2 x = \tfrac{1}{2}(1 - \cos 2x)$ · $\tan^2 x = \sec^2 x - 1$

3 — Is the derivative of an "inside" function present?

$\displaystyle\int\frac{\mathrm{f}'(x)}{\mathrm{f}(x)}\,dx = \ln|\mathrm{f}(x)| + c$ · $\displaystyle\int\mathrm{f}'(x)[\mathrm{f}(x)]^n\,dx = \frac{[\mathrm{f}(x)]^{n+1}}{n+1} + c$ — or substitute

4 — Is it a product of unrelated functions?

By parts: $\displaystyle\int u\frac{dv}{dx}\,dx = uv - \int v\frac{du}{dx}\,dx$

5 — Is it a fraction with a factorising denominator?

Partial fractions, then logs

Worked Example — Five integrals, five methods

Find (a) $\int xe^{x^2}\,dx$ (b) $\int xe^x\,dx$ (c) $\int\dfrac{e^x}{e^x+1}\,dx$ (d) $\displaystyle\int_3^4\frac{4}{x^2-4}\,dx$ (e) $\displaystyle\int_0^{\pi}\sin^2 x\,dx$.

①(a) $2x$ is the derivative of $x^2$ and $x$ is half of it — recognition: $\tfrac{1}{2}e^{x^2} + c$.

②(b) No inside derivative present; a product — by parts with $u = x$: $xe^x - e^x + c$.

③(c) The numerator is the derivative of the denominator: $\ln(e^x + 1) + c$.

④(d) $\dfrac{4}{(x-2)(x+2)} \equiv \dfrac{1}{x-2} - \dfrac{1}{x+2}$, so $\Big[\ln\dfrac{x-2}{x+2}\Big]_3^4 = \ln\tfrac{1}{3} - \ln\tfrac{1}{5} = \ln\tfrac{5}{3}$.

⑤(e) Identity: $\displaystyle\int_0^{\pi}\tfrac{1}{2}(1 - \cos 2x)\,dx = \Big[\tfrac{1}{2}x - \tfrac{1}{4}\sin 2x\Big]_0^{\pi} = \tfrac{\pi}{2}$.

(a) and (b) look almost identical, but one is a chain-rule pattern and the other a product. Asking "is the derivative of the inside sitting outside?" separates them instantly.

Every part of DfE content area H (Integration) in the second year, and the page that covers it.

HStandard Integrals$e^x$, $\tfrac{1}{x}$, trig functions; $\mathrm{f}(ax+b)$, $\tfrac{\mathrm{f}'}{\mathrm{f}}$, $\mathrm{f}'\mathrm{f}^n$; using trig identities.
HIntegration by SubstitutionChoosing a substitution, replacing $dx$, and changing the limits.
HIntegration by PartsThe formula, choosing $u$, repeated parts and $\int\ln x\,dx$.
HIntegrating with Partial FractionsDistinct and repeated linear factors, and improper fractions.
HAreas and Integration as a LimitThe limit of a sum, areas between curves and under parametric curves.
HDifferential EquationsSeparating variables, general and particular solutions, and modelling.

Numerical integration with the trapezium rule belongs to content area I and is on the Numerical Methods pages.

Multiplying instead of dividing. $\int e^{3x}\,dx = \tfrac{1}{3}e^{3x} + c$, not $3e^{3x}$. Differentiating multiplies by the inside's coefficient; integrating divides.

Dividing by a variable. $\int e^{x^2}\,dx$ is not $\dfrac{e^{x^2}}{2x}$. The reverse chain rule only adjusts by constants.

Wrong sign on trig. $\int\sin x\,dx = -\cos x + c$. Remember: differentiating $\cos$ gives a minus, so integrating $\sin$ does too.

Treating $\sin^2 x$ as a power. $\int\sin^2 x\,dx \neq \tfrac{1}{3}\sin^3 x$. Use the double-angle identity.

Keeping $x$-limits after substituting. Once the integral is in $u$, the limits must be $u$-values.

Losing the constant in a differential equation. Add $+\,c$ as soon as you integrate, and find it before exponentiating: $\ln y = x + c$ gives $y = Ae^x$, not $e^x + c$.

Standard

$\int e^x = e^x$, $\int\tfrac{1}{x} = \ln|x|$, $\int\cos x = \sin x$, $\int\sin x = -\cos x$, $\int\sec^2 x = \tan x$.

Linear inside

Divide by the coefficient of $x$.

f′/f

$\ln|\mathrm{f}(x)| + c$

Substitution

Replace $dx$ and the limits.

Parts

$uv - \int v\,u'\,dx$; $u$ = $\ln x$ or the polynomial.

Partial fractions

Split, then each piece is a log or a power.

Between curves

$\int(\text{top} - \text{bottom})\,dx$

Parametric area

$\int y\,\dfrac{dx}{dt}\,dt$

Separable DE

$\int\tfrac{1}{\mathrm{g}(y)}\,dy = \int\mathrm{f}(x)\,dx$

Ten questions drawing on the whole Integration topic. Decide on the method before you start writing.

Question 1

Find $\displaystyle\int\left(e^{2x} + \frac{1}{x}\right)dx$.

▶ Show solution

$= \tfrac{1}{2}e^{2x} + \ln|x| + c$

Question 2

Find $\displaystyle\int\cos(3x + 1)\,dx$.

▶ Show solution

Linear inside, so divide by $3$: $\tfrac{1}{3}\sin(3x+1) + c$.

Question 3

Find $\displaystyle\int\frac{x}{x^2+4}\,dx$.

▶ Show solution

$\dfrac{d}{dx}(x^2 + 4) = 2x$, and we have half of it.

$= \tfrac{1}{2}\ln(x^2 + 4) + c$ (no modulus needed since $x^2 + 4 > 0$).

Question 4

Find $\displaystyle\int\cos x\sin^4 x\,dx$.

▶ Show solution

$\cos x$ is the derivative of $\sin x$, so this is $\mathrm{f}'(x)[\mathrm{f}(x)]^4$.

$= \tfrac{1}{5}\sin^5 x + c$

Question 5

Find $\displaystyle\int xe^{3x}\,dx$.

▶ Show solution

By parts: $u = x$, $\dfrac{dv}{dx} = e^{3x} \Rightarrow v = \tfrac{1}{3}e^{3x}$.

$= \tfrac{1}{3}xe^{3x} - \displaystyle\int\tfrac{1}{3}e^{3x}\,dx = \tfrac{1}{3}xe^{3x} - \tfrac{1}{9}e^{3x} + c$

Question 6

Find $\displaystyle\int\frac{1}{x(x+2)}\,dx$ for $x > 0$.

▶ Show solution

$\dfrac{1}{x(x+2)} \equiv \dfrac{1/2}{x} - \dfrac{1/2}{x+2}$

$= \tfrac{1}{2}\ln x - \tfrac{1}{2}\ln(x+2) + c = \tfrac{1}{2}\ln\dfrac{x}{x+2} + c$

Question 7

Use the substitution $u = 1 - x$ to evaluate $\displaystyle\int_0^1 x(1-x)^4\,dx$.

▶ Show solution

$x = 1 - u$, $dx = -du$. Limits: $x = 0 \Rightarrow u = 1$; $x = 1 \Rightarrow u = 0$.

$\displaystyle\int_1^0(1-u)u^4(-du) = \int_0^1\left(u^4 - u^5\right)du = \tfrac{1}{5} - \tfrac{1}{6} = \tfrac{1}{30}$

Question 8

Find the area enclosed between $y = x^2$ and $y = 2x$.

▶ Show solution

$x^2 = 2x \Rightarrow x = 0$ or $2$. The line is on top between them.

$\displaystyle\int_0^2(2x - x^2)\,dx = \Big[x^2 - \tfrac{1}{3}x^3\Big]_0^2 = 4 - \tfrac{8}{3} = \tfrac{4}{3}$

Question 9

Solve $\dfrac{dy}{dx} = xy$, given $y = 2$ when $x = 0$.

▶ Show solution

$\displaystyle\int\frac{1}{y}\,dy = \int x\,dx \Rightarrow \ln y = \tfrac{1}{2}x^2 + c$.

$x = 0$, $y = 2$: $c = \ln 2$. So $y = e^{x^2/2 + \ln 2} = 2e^{x^2/2}$.

Question 10

(a) Show that $\displaystyle\int_1^e\ln x\,dx = 1$. (b) Find the exact value of $\displaystyle\int_0^{\pi}x\cos x\,dx$. (c) A student says "the answer to (b) is negative, so the question is wrong". Explain why the student is mistaken.

▶ Show solution

(a) By parts with $u = \ln x$, $\dfrac{dv}{dx} = 1$: $\int\ln x\,dx = x\ln x - x$.

$\Big[x\ln x - x\Big]_1^e = (e - e) - (0 - 1) = 1$ ✓

(b) $u = x$, $v = \sin x$: $\displaystyle\int x\cos x\,dx = x\sin x - \int\sin x\,dx = x\sin x + \cos x$.

$\Big[x\sin x + \cos x\Big]_0^{\pi} = (0 - 1) - (0 + 1) = -2$

(c) A definite integral is a signed area. For $\tfrac{\pi}{2} < x < \pi$, $x\cos x < 0$, and that part of the curve below the axis outweighs the part above it, so a negative value is perfectly possible.